Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}$, $\frac{\pi}{8}$ and $\frac{\pi}{4}$. | x | 0 | $\frac{\pi}{16}$ | $\frac{\pi}{8}$ | $\frac{3\pi}{16}$ | $\frac{\pi}{4}$ | |--------------|------------|------------------|------------------|-------------------|------------------| | y | 1 | 1.01959 | 1.08239 | 1.20269 | 1.41421 | The exact value of $\int \sec \, x \, dx$ is $\ln (1 + \sqrt{2})$. Calculate the % error in using the estimate you obtained in part (b).

A-Level Edexcel Maths Pure Polynomials: Given that $y = ext{sec} \, x$,

Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}$, $\frac{\pi}{8}$ and $\frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 7

Question 5

$Given-that-$y-=--ext{sec}-\,-x$,-complete-the-table-with-the-values-of-$y$-corresponding-to-$x-=-\frac{\pi}{16}$,-$\frac{\pi}{8}$-and-$\frac{\pi}{4}$-Edexcel-A-Level Maths Pure-Question 5-2006-Paper 7.png$

Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}$, $\frac{\pi}{8}$ and $\frac{\pi}{4}$. | x ... show full transcript

Worked Solution & Example Answer:Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}$, $\frac{\pi}{8}$ and $\frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 7

Step 1

Complete the table

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Answer

To find the values of $y = ext{sec} \, x$ for the given $x$ values:

For $x = \frac{\pi}{16}$ :

$y = \text{sec} \left( \frac{\pi}{16} \right) \approx 1.01959$
For $x = \frac{\pi}{8}$ :

$y = \text{sec} \left( \frac{\pi}{8} \right) \approx 1.08239$
For $x = \frac{3\pi}{16}$ :

$y = \text{sec} \left( \frac{3\pi}{16} \right) \approx 1.20269$
For $x = \frac{\pi}{4}$ :

$y = \text{sec} \left( \frac{\pi}{4} \right) \approx 1.41421$

Step 2

Use the trapezium rule

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Answer

To estimate $\int \sec \, x \, dx$ using the trapezium rule:

[ I \approx \frac{1}{2} \times \frac{\pi}{16} \left(1 + 1.01959 + 1.08239 + 1.20269 + 1.41421 \right)
]

[
= \frac{\pi}{32} \left(9.02355 \right) \approx 0.8859
]

Step 3

Calculate the % error

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Answer

The exact value of the integral is $\ln(1 + \sqrt{2}) \approx 0.88137$ .

To calculate the percentage error:

[
\text{Percentage error} = \frac{|\text{approx} - \text{exact}|}{\text{exact}} \times 100
]
Substituting in the values gives:

[
\text{Percentage error} = \frac{|0.8859 - 0.88137|}{0.88137} \times 100 \approx 0.51 %
]

Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}$, $\frac{\pi}{8}$ and $\frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 7

Worked Solution & Example Answer:Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}$, $\frac{\pi}{8}$ and $\frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 7

Complete the table

Use the trapezium rule

Calculate the % error

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